Optimal nonequilibrium thermometry in Markovian environments

Pavel Sekatski and Martí Perarnau-Llobet

Department of Applied Physics, University of Geneva, 1211 Geneva, Switzerland

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What is the minimum time required to take a temperature? In this paper, we solve this question for a large class of processes where temperature is inferred by measuring a probe (the thermometer) weakly coupled to the sample of interest, so that the probe's evolution is well described by a quantum Markovian master equation. Considering the most general control strategy on the probe (adaptive measurements, arbitrary control on the probe's state and Hamiltonian), we provide bounds on the achievable measurement precision in a finite amount of time, and show that in many scenarios these fundamental limits can be saturated with a relatively simple experiment. We find that for a general class of sample-probe interactions the scaling of the measurement uncertainty is inversely proportional to the time of the process, a shot-noise like behaviour that arises due to the dissipative nature of thermometry. As a side result, we show that the Lamb shift induced by the probe-sample interaction can play a relevant role in thermometry, allowing for finite measurement resolution in the low-temperature regime. More precisely, the measurement uncertainty decays polynomially with the temperature as $T\rightarrow 0$, in contrast to the usual exponential decay with $T^{-1}$. We illustrate these general results for (i) a qubit probe interacting with a bosonic sample, where the role of the Lamb shift is highlighted, and (ii) a collective superradiant coupling between a $N$-qubit probe and a sample, which enables a quadratic decay with $N$ of the measurement uncertainty.

Thermometry is a basic metrological task that is vital throughout science and technology. Estimating temperature is important on all scales, ranging from astronomical bodies with temperatures in the millions of kelvins to atomic systems near absolute zero. In particular, applications of thermometry in nanoscale or microscale devices are becoming increasingly relevant as technology advances. It is therefore crucial to understand the fundamental limits for temperature estimation in quantum systems, and how to approach them in practice.

Here, we establish such limits whenever the sensing time is limited. In other words, we address the question: what is the minimum time required to take the temperature of a sample with a given precision? To address this question, we consider a probe (thermometer) interacting with the sample (at some temperature) for a finite time. We then derive fundamental bounds on the measurement precision, and show that in many scenarios these limits can be saturated with a relatively simple experiment.

These limits are valid in a broader context where a physical property of a sample is estimated through its interaction with a probe, whose evolution is well described by a Markovian evolution. Indeed, our results can be understood as a speed limit relating a particular sample-probe interaction with the amount of information (quantified by the Quantum Fisher Information) that can be acquired by the probe in a finite amount of time. Our results hence provide a general framework for placing fundamental limits in finite-time sensing in open (quantum) systems, as well as practical strategies to saturate these bounds.

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